Apparatus for improving the homogeneity of a magnetic field



Aug. 3, 1965 w. A. ANDERSON 3,199,021

APPARATUS FOR IMPROVING THE HOMOGENEITY OF A MAGNETIC FIELD Aug. 3, 1965 w. A. ANDERSON 3,199,021

APPARATUS FOR IMPROVING THE HOMOGENEITY OF A MAGNETIC FIELD IN V EN TOR. Wso n Miniera-arr ero/2, Trove/Viv Aug. 3, 1965 w. A. ANDERsoN 3,199,021

APPARATUS EoR IMPRovING THE HoMoGENEITY 0E A MAGNETIC FIELD Filed Dec. 19, 1960 4 Sheets-Sheet 5 FIEH9 ug- 3, 1965 w. A. ANDERSON 3,199,021

APPARATUS FOR IMPROVING THE HOMOGENEITY OF A MAGNETIC FIELD Filed Deo. 19, 1960 4 Sheets-Sheet 4 INVENTOR.

aies nai n :29 dependentiy removing certain iir-.ear line broadening components ofthe polarizing magnetic geid,

FIG. 9 depicts a rectilinear coil geometry for removing undesired line broadening magnetic field components,

FIG. 10 shows the rectilinear coil configuration for independently removing certain other quadratic line broadening components of the polarizing magnetic field,

FIG. 11 shows an isometric view of the curvilinear coil configuration for removing the linear line broadening components of the polarizing field in the direction of the polarizing field,

FIG. 12 shows a schematic isometric View of the cur'- vilinear coil configuration for removing the linear line broadening components of the polarizing magnetic field along the Y axis,

FIG. 13 shows a schematic isometric view of the curvilinear coil configuration for removing certain quadratic line broadening components of the polarizing magnetic field,

FIG. 14 shows the graphic solution of the parameters of the coil configuration of FIG. 13,

FIG. l shows the graphic solution of the ampere turn relationships for the coil configuration of FIG. 13,

FIG. 16 shows a schematic isometric view of the curvilinear coil configuration for removing certain quad'- ratic line broadening components of the polarizing magnetic field, and

FIG. 17 shows an arrangement of four different coil configurations for removing all linear line broadening components and certain quadratic line broadening components of the polarizing magnetic field.

Referring now to FIG. 1 there is shown in schematic block diagram form a typical high resolution nuclear resonance spectrometer useful for obtaining high resolution chemical spectra employed in chemical analysis and the like. More particularly, a substance which it is desired to analyze is placed in a suitable container such as, for example, a glass vial and immersed in a strong polarizing magnetic field such as that produced between the pole pieces of a powerful electromagnet 2.

' An RF. transmitter 3 supplies an alternating high frequency magnetic field to the sample, the time varying RF. magnetic field being oriented approximately at right angles to the D.C. polarizing magnetic field produced by magnet 2. A receiver coil 5 is closely coupled to the sample of matter under analysis and is oriented approximately at right angles to the transmitter coil 4 and D.C. polarizing magnetic field such as to minimize the direct coupling between the RF. transmitter 3 and the receiver coil 5 and to obtain maximum coupling to the precessing gyromagnetic bodies making up the sample of matter under analysis. The receiver coil 5 is coupled to a preamplifier 6 which in turn feeds an RF. amplifier 7. The output of the RF. amplifier 7 is detected in detector 8 and thence amplified in D.C. amplifier 9 and the output thereof is fed preferably to the vertical axis of oscilioscope l@ and/or to the vertical axis of a grap-hic recorder 11.

A sweep generator 12 supplies a sweep current to a pair of aiding connected sweep coils 13 disposed straddling the sample under analysis for sweeping the total polarizing magnetic field through resonance of the sample under analysis such that a gyromagnetic resonance spectrum of the sample may be obtained on the oscilloscope and/ or graphic recorder 11.

A pair of bucking connected electric shim coils 14 are disposed straddling the sample of matter under analysis and are supplied with curent from a battery 15 via switch 16 and variable resistor 17.

In operation, while observing resolution of the gyromagnetic resonance spectrum on the oscilloscope or graphic recorder, switch 1.6 is closed and variable resistor 17 is adjusted for optimum resolution.

Although only one set of electric shim coils has been depicted in FIG. 1 in a practical application of electric shimming a plurality of independent electric shim coils the sample volume.

A. will be utilized. Particular geometries and disposition of shim coils for preferred embodiments of the present invention will be shown and described below.

Electric shim coils 14 may be carried upon the gyromagnetic resonance probe, not shown, which normally houses the sample vial 1, transmitter coil 4, detector coil S, and sweep coils 13 and is positioned in the gap of the magnet. However, in a preferred embodiment in the present invention, electric shim coils 14 are embedded in dielectrical thermally insulating pole cap covers fixedly secured to the mutually opposing pole faces of the electromagnet. .By passing a separate controllable electric current through each of the electric shim coils f4, small local fields of corrective magnitude and direction may 3 be set up in the magnet air gap to counteract remaining eld gradients over the sample dimensions for obtaining optimum gyromagnetic resonance spectra resolution and/ or removing spinning sidebands.

The particular electric shim coil geometries of this invention will be derived in detail below. However, briefiy stated, this derivation calls for resolving the magnetic field into its possible components making a first simplified assumption that in the magnetic field region of interest there are no electric currents or ferromagnetic bodies, allowing one to expand the magnetic field as the gradient of a scalar potential. This results in a substantial reduction in the number of magnetic field components that must be considered and reduces these components to a certain number of independent components which can be seen in Table I below.

A second simplifying assumption is made that one of the D.C. components of the magnetic field is much, much greater than the other possible two over the sample volume and this reduces the number of magnetic field components which have to be considered.

A third simplifying assumption is made that the sample will be spun to average out magnetic field components that are not along the axis of spinning and this further simplifies the problem resulting in only having to con sider those components of the field in the direction of the spinning axis of the sample.

Some additional components of the field are considered to remove magnetic field components that are usually averaged out by sample spinning. This allows removal of spinning sidebands and also improves spectra resolution where the sample cannot be easily spun. The result is that from a total number of magnetic field components of 363, through fourth order field components, the simpliying assumptions allow control of the field for most practical purposes by the use of six independent control knobs.

A detailed derivation of the particular electric shim coil configurations and dispositions will now be made. Since the polarizing magnetic field is a vector field it can be resolved into three DC. components Bx, By, and BZ a component along each direction in S-dimensicnal space. Since it is possible for each of these three primary components to have linear gradients over the sample volurne in the x direction, y direction and z direction, this creates the possibility of a total of 9 linear gradients over Furthermore, since the linear or rst order gradients can also have gradients in three directions over the sample volume this creates a total of 27 possible quadratic gradients. Furthermore, since each of the quadratic gradients can have a third order gradient in any one of the three directions, there can be a total of Sl third order gradients. Since each of the third order gradients can lead to a fourth order gradient along any one of the three directions over the sample volume this creates a total of 243 possible fourth order gradients. Thus, the total number of magnetic field components up to and including all possible fourth order gradients is 363. The total number of filed derivatives is shown in column 2 of Table I below.

It is immediately apparent that the use of independent shim coils to remove each one of the possible gradients tabulated in column 2 of the table above would result in a prohibitive number of electric shim coils.

A number of simplifications will, therefore, be used in order to reduce the number of shim coils necessary.

An important requirement is that the shims be orthogonal so that controls will be independent of each other. This can be achieved by considering the constant terms in the spherical harmonic expansion of the magnetic field within the magnet gap.

Choosing the center of the sample as the origin of our coordinates and expanding by scalar potential I at a point P about this origin leads to the well-known spherical harmonics where, referring to FlG. 2, gb is the scalar potential at a point P, P111 is the Legendre polynomial of order m and degree n, Anm and Bum are constants: r is the distance between the origin and point P, 6 is the angle between r and the z axis and 1r is the angle between the x axis and the projection of r on the xy plane.

Since the expansion will satisfy the Laplace equation, `72iv=0, each of the constant terms thereof is uniquely determined for given field conditions. Therefore, expansion (l) immediately yields the number of independent components in each order of a field derivative. A few examples will clarify this statement. With 11:1, the expansion (l) represents the three components of the field;

With 11:2, the expansion (l) represents the linear field gradients. Although there are nine linear field gradients, only five of these are independent. Writing them in both spherical and rectangular coordinates they take the following forms:

By resigning shim coils which correspond to the various terms of such an expansion one may obtain a set of electric shims having controls which are substantially independent. The number of gradients to be considered is therefore reduced again as shown in column 3 of Table I.

The second simplifying assumption is thatone of the D.C. components of the magnetic field is much, much greater than the other possible two over the sample volume. Let the strong magnetic field B be chosen in the Z direction of FIG. 2 and let the small variations in the field due to inhomogeneity be B'. The total field is thus:

The variation in the resonance frequency of the sample 6 being tested is determined by the variation in B over the sample:

B has been found to be less than 10-1 gauss. Thus, we may expand with the result Here we see that the X and Y components of B contribute only in the order of l06 gauss to the field variation and therefore they may be neglected. Thus, it is only necessary to consider the field components parallel to the Z-axis. Since BX and By need not be considered and since BZ also satisfies the Laplace equation we may express BZ directly in terms of the spherical harmonics where the Anm and Bnm terms are constants with n representing the degree of the Legendre polynomial and m representing the order of the term.

It is seen from this expression that we have only one component of BZ for the first degree of Legendre polynomial, three components of BZ for the second degree of Legendre polynomial, five components of BZ for the third degree of Legendre polynomial, etc. This result is tabulated in column 4 of Table I. Thus, from our original total of 353 field gradients in the first ve degrees of Legendre polynomials, We now have only 25 components of BZ which must be controlled in order to substantially eliminate field gradients in the vicinity of our sample.

Spinning the sample will result in a further reduction of necessary controls. The effect of sample spinning is to average out all magnetic field gradients except those in the axis of spin. Assuming spin about the Y axis, all the terms of Equation 7 which include forms of X and Z will be averaged out and therefore need not be considered further. The remaining terms of Equation 7 affect spinning resolution of the sample where spinning resolution is the resolution of the resonance lines in a gyromagnetic resonance spectrum obtained from a spinning sample` The A10 term of these corresponds to the uniform field BZ and can be disregarded. The remaining terms in Equation 7 which correspond to field gradients which must be considered lfurther are The results of sample spinning and the type of field gradients remaining are tabulated in columns 5 and 6 of Table I.

Equations corresponding to certain field gradients Will now be derived. These equations will be used to design shim coils for controlling the eld gradients which influence spinning resolution.

'7 In FIG. 2, 20 represents a linear portion` of coil I8 approximating for purposes of derivation an infinitely long wire in the X direction. The magnetic field produced by this wire at any point in the coordinate system is represented by the following expression:

Bziyilsz) *m (8) where /io is the permeability of free space, Ix is the current in the wire, Y0 is the distance to the wire from the Z-aXis in a y direction, and Z0 is the distance to the wire along the Z-axis. Although the use of an equation for the magnetic field about a iinite wire will yield more exact results, the Equation 8 is of suicient accuracy for the purposes of gyromagnetic resonance where diameters of the active portion of sample l are in the order of fractions o centimeters While the linear portion 2() of coil 18 is in the order of l5 more centimeters of length.

The magnetic field gradients at the origin, for a long straight wire, are readily calculated from lEquation 8. With R=Y0/Z0, the eld gradients evaluated at the point x=y=z=0 have the following form:

@MBL-NBL @SBZ 01. einti-ioRZ/sJfRe The symbols Y0, Z0 and R will relate to the same parameters mentioned above whenever referred to in the derivations of each coil to follow.

Before considering specific current configurations it isY useful to consider the atiect of the symmetry of the current distribution upon the various field gradients.

The magnetic iield B produced by a current density j is given by the Biot and Savart law Here j is the current per unit area, lu0=permeability of free space,

r=coordinates of field point,

. r=coordinates of current source, dv=increment of volume of current source.

The geometries which are most applicable to most magnets are those in which the current distribution is confined to two parallel sheets at Z=Z0, one against each magnet pole face 2 as shown in FIGURE 2. The volume integral may then be reduced to two surface integrals which have the form (2l) Here ]X(x, y', izo), Jy(x', y', izo) are the current densities per unit length on the surfaces at z=iz0. The total field and any of its derivatives may be calculated from (2l). Let the point of measurement which corresponds to the center of the sample be at the position x=y=Z=0. Inspection of (2l) shows that if JX and Jy are identical at iZO then at the origin the Ziield BZ and all derivatives with respect to either x and y are identical for the two current sheets with the result that the total gradients of this type are just twice that of a single current sheet. Gradients of the form I- with n an odd integer have opposite signs and thus the total gradient of this type vanishes at the origin.

On the other hand if the conductor configuration on the two sheets is identical but the currents iiow in opposite directions, then Bz vanishes in the plane z=0 and thus all gradients of BZ with respect to x or y also vanish in this plane as Well as gradients of the type with n even.

Generally the conductor configuration will be symmetric upon reilection in the x-z and y-z planes. The current densities IX and Jy will then differ at most in sign at the points ixiy. In addition the symmetry of one current component will determine that of the other component through the requirement that the divergence of the current density is zero.

If Mx, y)=Jy(-x, y) the Mx, y)= X(-x, y) and inspection of (2l) shows that the eld BZ vanishes in the y plane x20 and thus all the derivatives of Bz with respect to y or z must vanish as well as the even derivatives of BZ with respect to x. It, however,

Mx, y)=-Jy(x, y) then ]X(x, y)=]X(-x, y) and all odd order gradients of BZ with respect to x will vanish.

Similarly if ]X)x, y) =]X(x, -y), then Jy(x1 y)=*]y(xr BZ will vanish in the x plane y=0 and thus all derivaow and a compact structure. The symmetry considera-Y DI tions above require proper current flow in the longer coil sides of FIG. 2 but do not require any particular connection between these coil sides. This is also true of all the rectangular coils described hereafter.

Rectangular coil geometries are derived below for removing the eld gradients which atect spinning resolution. ln deriving these coils, Equations 9-l9 which correspond to certain gradients. will be utilized.

Gnly the currents in coil sides located in the upper half of the plane Z: -i-ZO as seen in FTG. 2 will be considered explicitly as the currents in coil sides located in the plane Z:-Z and those in the lower half of plane Z:i-Z0 are located by the symmetry considerations discussed above. Use is also made of coordinate transformations that correspond to rotation about the Z `axis.

FIG. 2 shows coil geometry which independently removes gradient corresponding to term B21 of Equation 7. A pair` of rectangular coils 13 and 19 are collateral along their length in a first plane adjacent one of the pole pieces 2. A second pair of rectangular coils 18 and 19 are located opposite the first pair of coils in a second plane adjacent the other pole piece 2. Each of the coils 1S and 19 have the same dimensions including a width (a) and the centers of coils 18 and 19 in each plane are a distance (ZYC) apart. The planes containin the rectangular coils 18 and 19 are parallel and separated by a distance (220). The longer sides 2t) of the rectangular coils 18 and 19 form linear current paths parallel to the plane 32:0 and are spaced apart by a `distance large compared to their cross section. The coils 1S and 19 are connected in series in a manner to provide the same direction of current ow in each of the coils 1S and a direction of current How in each of the coils 19 opposite to that in coils 1S. The parallel connected batteries 24 supply current through the variable resistance 25. Both of the coils 18 are above the plane y:0 in the coordinate system shown and both the coils 19 are below the plane 31:0. Thus, with JX(x,y):-|]X(x, y), BZ will vanish in the plane y:0. An examination of the coil symmetry discussed above will show that all of the gradients expressed in Equations. 9 through 19, except 10, 14 and 18 will vanish because of symmetry. Equating the expression 14 to ero will yield a location for the coil sides which will cause the @SBZ term to vanish. Solving this gives the values R=i (\/2-{l) and R: |:(\/2-1). Now, with the current directions shown in FIG. 2 and using the positive values for R and R found above, the contributions to @BZ @y will add, giving this shim a resultant control of the en, @y

Cil

1i) however, all that is desired is that the gradient from the two legs cancel. Thus, for any coil of width a one may calculate the value of coil center Yc which will cause the gradient to vanish. The results of such a calculation have been plotted in FiG. 3, which shows Xe s i Z0 for the gradient It turns out that for each value of a there are two values of YC, Which cause the third order gradient to vanish. Of course, the value of the desired gradient es, @y

will also depend `upon the coil spacing a. This relationship is shown in FIG. 4 which is a graph showing the magnitudes of the desired gradient y VS. for the two possible values of YC. The dotted curve in FIG. 4 corresponds to the dotted curve in FIG. 3. It will be noted that the particuiar solutions for a and Yc found above will lie on the curves plotted in FGS. 3 and 4. The proper use of the curves shown in FIGS. 3 and 4 will given the dimensions for a coil shim which will independently control the en, @y

gradient.

The positioning of the coils 1S and 19 relative to the pole pieces 2, the current supplies 24 and the axes shown in FIG. 2 is a typical arrangement. It will be assumed in discussion of the remaining shim coils that they are positioned in the same Way. They will be shown alone, however, for purposes of simplicity and clarity.

The next two terms in Equation 7 which influence the spinning resolution `of the `sample are A30-(2Z2-X2-Y2) and A32(X2-Y2). Each of these terms correspond to a @y2 gradient. A simpler shim coil arrangement will result from a linear combination of these two terms yielding two new orthogonal terms which have the form Aao (z2-y2) and A32(z2-x2). This linear combination has the added advantage that only the A3(z2-y2) will now atiect the spinning resolution because of a gradient. It is therefore necessary to derive a shim in accordance with Equations 9 through 19 which will produce a gradient corresponding to this A30 term. To pro- Y duce such a gradient with gradients of the type e faces shall be identical in Symmetry will then cause with n odd to vanish in the plane z= and with Jx(x, y) =-Jx(x, y) all terms with n odd will vanish in the plane )1 0. With these conditions, it is obvious that a single current tilament cannot cause BZ to vanish and produce the desired gradient. The field BZ will vanish at the origin, however, if two iilaments having currents of equal magnitude but opposite directions are located at R and R where either R=R or R-l/R. The second of these solutions yields The parameter R should now be chosen at an optimum value. However, with the conditions expressed above, there is no value of R which will cause the term of Equation 16 to vanish while the desired gradient of Equation 12 remains finite.

The fourth order term to vanish. This is done by equating Equation 16 to Zero,

yielding the values R2=5|2\/ and (R)2=5-2\/5.

FIG. shows a coil arrangement for removing the gradient corresponding to the linear combination A term obtained from Equation 7. One pair of concentric rectangular coils 26 and 27 are located in a rst plane adjacent one pole piece. Another pair of concentric rectangular coils 26 and 27 are positioned opposite the first pair of rectangular coils 26 and 27 in a second plane adjacent the other pole piece. Each of the outer coils 26 of each pair have a width (ZYO) and each of the inner coils 27 of each pair have a width (ZYO). The planes containing coils 26 and 27 are parallel and are separated by a distance (220). The inner coils 2.7 are connected in series with a potentiometer 28 and this combination is connected in parallel with the series connected outer coils 26 in a manner to provide the same direction of current flow in each of the coils 26 and a direction of current flow in each of the coils 27 opposite to that in the coils 26. The current ratios in the two ilaments are then adjusted by means of potentiometer 28 so that the iield vanishes at the origin. This requires the proportion ViP-Mish ,/5) *1.617

where IN represents the ampere turns in the outer coils 26 and IN represents the ampere turns in the inner coils 27.

The above requirement that each current lament produce 4BZ gradient is not mandatory, however, and may he replaced by the less stringent demand that the total ff gradient for both filaments vanish at the origin. Thus,

for any value of R one can find a value of R and a ratio of currents that will result in both BZ and vanishing. The graph of FIG. 6 plots values of 2R1 and ZR', which cause BZ and to vanish at the origin. Given a particular value for R, this graph can be used to nd a proper value for R. Similarly, the graph of FIG. 7 plots values of ampere turn ratios vs. 2R which cause BZ and to vanish at the origin. The proper ampere turn ratio for a particular value of R may be determined by using this graph.

rhe last two terms in Equation 7 which influence the spinning resolution of the sample are 15B4Y(4Z2 Z2X2Y2)/2 and lO5B43l/(3X2-Y2). Each of these terms correspond to a gradient. A convenient linear combination of the B41 and B43 terms will yield a field of the type Such a rleld would be produced by identical current distributions at iZQ; and with .x(x, y)=-JX(x, y). A single coii side at R=1 would produce the desired gradient and no lower order terms. However, this would create the disadvantage that all currents ow in the same direction making it necessary to close the coil on the outside ofthe magnet.

A more convenient arrangement is achieved by locating two coil sides at positions R and R which have currents of equal magnitude but flowing in opposite directions. The general coniguration is the same as that for the gradient shown in FIG. 2 with the outer coil sides in each plane separated by a distance (ZYO) and the inner coil sides in each plane separated by a distance (2 0). r:The only gradients produced by this coil are the i3 will then vanish it Yo, Y'O and ZO are chosen so that R52-r3 (R R2-1 and the desired gradient @SBZ y will be a maximum with either R or R' as near to unity as possible. A shim for controlling the fourth order are eliminated by using two currents of unequal magnitude flowing in opposite directions in the upper halt of the planes ZziZo. Solving Equations 8 and l0 when adapted to this situation yields the following relationship between positions and magnitudes of the current filaments:

With these four shims an independent control is obtained of all the terms in Equation 7 through five degrees of Legendre Polynomial which will atleet the spinning resolution of the sample. In practice, however, certain other terms in Equation 7 influence the homogeneity of the magnetic field without atliecting the resolution of the spinning sample. This results from the introduction of spinning magnetic side bands which are brought about by the sample spinning. The spinning side band phenomena is discussed in an article by G, A. Williams and H. S. Gutowsky appearing in the Physical Review, 104, 278 (1956).

An interesting result in regard to these spinning side bands is found if Equation 7 is mathematically transformed by interchanging the Y and Z axes, Such a transformation yields a new expansion for BZ in which each degree of the expansion has one zero order term which introduces gradients aliecting the spinning resolution, two first order terms which introduce first side band resonances, two second order terms which introduce second side band resonances, etc. This relationship can be seen in the following table:

Table I1 Terms Resulting From in the transformed expansion above, the zero order terms AQUY and A30(2Y2-X2Z2) affect spinning resolution; the lirst order terms AglX, B21Z, A31(2X Y) and B31(2ZY) introduce first side band resonances; the second order terms A32(X2-Z2) and B32(2XZ) introduce second side band resonances, etc.

It has also been found that deriving shims for the control of at least the lower degree side band terms will substantially increase the homogeneity of the magnetic field. These shims are derived in the same general manner as those previously mentioned.

FG. 8 shows a coil arrangement for removing a gradient which corresponds to the ZAZZ term in Equation 7. This gradient will introduce objectionable rst side band resonances. A first rectangular coil le having a width (ZYO) is located in a first plane and a second rectangular coil 14 of width (ZYO) is positioned opposite the rst coil in a second plane parallel to the iirst plane and separated therefrom by a distance (2f/L0). The rectangular coils it are series connected in a manner to provide opposite directions of current flow in the two coils. The gradients produced by these coils at the location of the sample will be considered by reference to Equations 9 through 19. Because of symmetry about the Z-axis and the opposite direction of current flow in the two coils, all gradients of the type @SBZ gradient. The simple solution being R=1. The fifth order Equation 19 then represents the first undesirable gradient. A Taylors expansion can be used to evaluate this term so that its magnitude can be compared with that of the desired gradient With 12:1, this expression may be written:

2 2 72 L 4 Bam, n :'MNFdM-M 102 I 5y i) This expression reveals that even with a relatively large sample having a z/zU-1/i when z is the radius of the sample, the undesired gradient WEZ is less than 2% of the desired gradient. It is therefore evident that the shim coils shown in FIG. 8 will give a substantially independent control of the @BZ z gradient.

The A21X term of Equation 7 produces a sa 'dx gradient which also introduces first side band resonances. It is identical to the previously mentioned gradient except that it is rotated 90 about the Z-'lxis Thus, the shim for this term is identical to that shown in EiG. 2 but it must be oriented at right angles thereto.

A shim to produce the A32 term formed by the linear combination of theA and A32 terms of Equation 7 is identical to that for the A30 term which is shown in FIG. 5 except that it is rotated by 90 about the Z-axis.

FIG. 9 shows a coil arrangement for removing a 'yz gradient which corresponds to the B31 term in Equation 7. A rectangular coil 30 and a rectangular coil 31 are collateral along their lengths in a first plane. Rectangular coils 32 and 33 are positioned opposite the rectangular coils 30 and 31 respectively in a second plane which is parallel to the tirst plane and separated therefrom by a distance (2Z0). The outer coil sides in both planes are separated by a distance (2Y0) and the inner coil sides in both planes are separated by a distance (ZY). All of the rectangular coils 30, 31, 32 and 33 are connected in series in a manner to provide the same direction of current flow in the coils 30 and 33 and a direction of current flow of the coils 3l and 32 opposite to that in coils 30 and 33. This arrangement will cause 'dan terms in Equations 9 through 19 for n even and Bz in the plane 2:0 to vanish because of symmetry conditions. BZ will also vanish in the plane y=0 with JX(x, y)=]x(x, -y). Because of these symmetry conditions the first term after the desired gradient to have a value Will be the yz3 term of Equation 17. The dimensions which will also cause this term to vanish are found by equating Equation 17 to zero. This results in the values R2=1i2\/5/5. The coils 36, 31, 32 and 33 are thus dimensioned with FIG. 10 shows a coil arrangement for removing a EZBZ xy gradient Which corresponds to the B32 term of Equation 7. A rectangular coil 34 and a rectangular coil 35 are located in each of parallel planes adjacent each pole piece with the parallel planes being a distance (ZZQ) apart. Each of the coils 34 and 35 have the same dimensions which include a length which is at least twice as large as its Width (ZYO). The perpendicular coilsl 34 and 35 in each of the planes have the same center and are positioned opposite to the coils 34 and 35 respectively in the other plane. The coils are series connected in a manner to provide the same direction of current flow in each of the coils 34 and a direction of current ow in each of the coils 35 opposite to that in the coils 34.

The dimensions for the coils 34 and 35 are calculated by setting Equation 14 to zero yielding the values gradient and is therefore more desirable.

The use of all the shim coils described above will give a substantially independent control of all the first, second, third and fourth order gradients which affect the magnetic resolution of a spinning sample and will eliminate certain low order gradients which produce side band resonances.

lthough it is possible to derive shirns for the control of higher order gradients, such derivations become increasingly more dii'licult. Furthermore, the use of the coils described above in various combinations has been found to produce magnetic eld which are suiiiciently homogeneous for most practical purposes.

Shims have also been designed which utilize circular current loops. Shims of this type are claimed in copending divisional application Serial No. 441,829, filed March 22, 1965. The magnetic eld due to the current ilowing in a circular loop can be described in cylindrical coordinates using complete elliptic integrals. On page 271 of International Series in Pure and Applied Physics by William A. Smythe, the axial eld component of a loop located in the plane z\=Z0 and centered atx', y is given by:

wehe-zona [Hf i www2-Zug B #DI z adonde-Zon Where rzradius of loop,

i and E are complete elliptic integrals of the modula The derivatives from this expression maybe calculated and evaluated at z=0. With the substitution L ot- Z0 these derivatives take the following form:

Circular coil shims will now be derived for removing field gradients which correspond to certain terms in Equation 7. Equations 22-27 will be used in these derivations. The symbols r, p, and ot will relate to the parameters described above Wherever referred to in the derivation of the circular coils which follow.

FIG. 11 shows a circular coil for removing gradient of Equation 23. This gradient corresponds to the A20 term of Equation 7. A circular coil 40 is adjacent each pole piece in parallel planes separated by a distance 2Z0. The coils 4() have their centers on the z-aXis and have the same radius r. The coils 4) are 17 connected in a manner to provide them with opposite directions of current iiow. This arrangement will cause the gradients represented by Equations 22, 24, 25 and 26 to vanish because of symmetry if leaving only Equation 27 to be considered. The expansion of the magnetic eld from the coil in FIG. 11 is as Thus, for a z/Z=1A where z is the radius of the sample and an a=2/\/ the field variation of the undesired BZ z5 gradient is only about 1% of the field variation of the desired represented by the A2l and B21 terms of Equation 7 offers a more difficult problem since in these cases the eld an-d its derivatives must be evaluated at a point removed from the Z-axis. In an article of the Journal of Applied Physics, 18, 976 (1947), John P. Blewett has tabulated BZ as computed from Equation for a number of values of p/r and (Z-Z)/r. Using Blewetts tabulation for the ratio has been plotted. A combination graphical method and the method of finite diierences has been used to also plot the first three derivatives of BZ with respect to p/r. These graphs indicate that for the values p/ :E114 and 0.83 the term o3B z3 while the value of Bz y is still more than one-half of its maximum value. Thus,

a shim to produce the eBz by gradient can be made in accordance with the arrangement shown in FIG. l2 which shows a coil 41 adjacent each pole piece and a coil 42 in each of a pair of parallel planes separated by a distance (ZZ). Each of the coils 41 and 42 have a radius r and are positioned so that each of their centers is distance p from the center of the sample being tested. Lines joining the centers of coils 41 and the centers of coils 42 on opposite pole pieces are perpendicular to the planes in which the pole pieces lie. The coils 41 and 42 are connected in series in a manner to provide the same direction of current flow in each 18 ot the coils 41 and a direction of current ow in each of the coils 42 opposite to that in the coils 41.

A coil conguraton to produce the gradient which corresponds to lthe A21 term of Equation 7 is the same as that shown for the gradient of FIG. 12 except that Iit must be rotated by 90 about the Z-axis.

FIG. 13 shows a circular coil shim for removing the quadratic ygradients yz bz2 y2 which correspond to the A30 term of Equation 7. A coil 45 of radius r and a coil 46 of radius r are located in parallel planes separated by a distance (ZZO) and each have each of .their centers on the z-axis. The coils 45 and 46 4are connected in parallel so as to provide for the same direction of current flow in each of the coils 45 and a direction of current ow in each of the coils 46 opposite to that in coils 45. Symmetry will then eliminate the gradients represented by each Equations 23 through 27 except .the desired gradient of Equation 24 and the fourth order term of Equation 26. The dimensions can then be determined which will cause the fourth order gradient to vanish by setting Equation 26 equal to zero, yielding the values of a2=6-{-2\/7 and (a)2=6-2\/7. A suitable configuration is .then obtained by dimensioning the coils 45 according to the positive value of (a2) above and the coils 46 according to the positive value of (a)2 above. The potentiometer 47 is connected in ser-ies With coils 46 and provides for adjustment of current in the inner coil to obtain the proper ampere turn ratio. These loops require a current ratio of I rgano where NI represents the ampere turns in coils 45 and NI' represents the ampere turns in coils 46.

A more general solution may be determined it the restriction that each separate loop must cause no fourth order gradient is replaced by the condition that any fourth order gradient lproduced by one loop is cancelled by the other loop and that the total field BZ vanish at the origin. Adding `a second term representing a second cur-rent loop to Equations 22 and 26 'and equating each to zero yields lthe desired solution. The resul-ts of this solution are shown graphically in FIGS. 14 and 15. In FIG. 14 the values of 1"/Z0 vs. r/Zu for are plotted. The values of r and r for a given Z0 which will eliminate the r and r which will maximize the desired gradient are equal or nearly equal to the values determined above, i.e.,

19 for a given Zo which will cause a cancellation ofthe 84Bz @zt gradient.

The next two gradients which affect spinning resolution of the sample will now be considered. The first of these is the gradient. A shim having coils in which Zeh-:0.36 and values of p/r=().55 and p/r=1.40 will introduce the 63BZ 6373 gradient and no 'linear gradient. The general current configuration is identical with the shim used to produce the 'BZ @y gradient which is shown in FIG. l2. Finally, the shim to produce a gradient may be constructed with the same general configuration las that which produced the gradient. This coil arrangement is shown in FIG. 13. i The proper dimensions for these coils can be determined by adding an additional term to Equations 22 and 24 to represent the second loop. These equations are then set equal to zero, yielding the relationship 4a2-l-9 f 2: (a 2 4 The current ratios may then be calculated from the relationship M ML M NI"u+ a' 2r/2 a2 FIG. 16 shows a circular coil arrangement for removing a `gradient corresponding to the B32 term of Equation 7. A pair of circular coils 50 and a pair of circular coils 51 are located in each of a pair of parallel planes adjacent the two pole pieces. The coils 50 in each parallel plane are positioned in the first and third quadrants of the x-y plane and the coils 51 in each parallel plane are positioned in the second and fourth quadrants of the x-y plane. The coils 50 and S1 are symmetrical about the center of the spinning sample and are connected in a manner to prov-ide the same direction of current flow in each coil 50 and a direction of current in each coil 51 opposite to that in the coils 50. This coil arrangement will give substan- .tially independent control of the gradient.

The rectangular and circular coils described above can be used in various combinations depending upon the type of field desired. Once a selection of coils has been made the coils may be mounted on a pair of compact shim structures for use immedately adjacent the pole pieces which produce the magnetic field.l Such a structure is shown in FIG. 17, for example. This shim 52 is made up of the four sets of coils 53. The coils 53 are the same as those shown in FIGS. 2, 11 and 13. A proper selection of currents in two such shims 52 mounted on magnetic pole pieces in addition to spinning of the sample between the shims will substantially remove gradients from the volume of magnetic field between the shims which heretofore would have required eight separate orthogonal coil sets on each pole piece. It will be noted in the shim arrangement of FIG. 17 that there are never more than two thicknesses of coil at any particular location on the shim. This allows the stacking of four separate gradient removing coils within the width of two coils since the flexible coils can be set flush against the supporting shim except where they are actually in contact with one another. A pair of such shims may then be mounted within an extremely narrow magnetic gap.

In the preceding description several conductor patterns have been set forth which will independently remove particular undesired field gradients from a volume of magnetic fields. It is readily apparent that numerous other conductor patterns may be employed for removing other field gradients as taught herein.

Since many changes could be made in the above construction and many apparently widely different embodiment of this invention could be made without departing from the scope thereof, it is intended that all matter contained in the above description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.

What is claimed is:

1. Apparatus for improving the homogeneity of a certain volume of magnetic field by removing a particular undesired field gradient including, electrically conductive means for providing a plurality of spaced apart current paths in each of a pair of essentially parallel planes, the spacing between said current paths being larger than any distance traversing their cross section, said pair of planes positioned on opposite sides of the volume of magnetic field to be corrected so as to define a gap therebetween, said electrically conductive means in each of said planes being thin in the direction taken across the gap so that said spaced apart current paths lie entirely in said pair of essentially parallel planes and lying on opposite sides of a plane through the volume of magnetic field and perpendicular to said parallel planes, said electrically conductive means being connected so as to provide when energized in each of said parallel planes two opposing directions of current flow on one side of said plane through said volume of magnetic field and two opposing directions of current flow on the other side of said plane through said volume of magnetic field, and said electrically conductive means being disposed so that when energized the currents flowing in opposing directions will flow substantially parallel to said plane through said volume of magnetic field.

2. Apparatus according to claim 1 including, means for spinning a gyromagnetic sample within the volume of magnetic field.

3. Apparatus according to claim 1 wherein, said plurality of spaced apart current paths are positioned so as to be substantially symmetrical about the center of the volume of magnetic Ifield.

4. Apparatus according to claim 1 including, for supplying a4 variable direct current to said apart current path means, and means for detecting field homogeneity within the certain volume of magnetic field for sensing maximum field homogeneity.

5. Apparatus for improving the homogeneity of a certain volume of magnetic field by removing a particular undesired gradient including a rst and a second pair of substantially rectangular concentric coils, all of said rec;

means spaced tangular coils each having a length which is at least twice as large as its width, said first pair of coils lying in a first plane, said second pair of coils lying opposite said rst pair of coils in `a second plane parallel to said first plane and separated by a distance (ZZU) therefrom, said rst and second pairs of coils each including an outer coil of width (ZYO) and an inner coil of width (ZYO'), means for providing each of said coils of width (2Y0) with ampere turns equal to IN) I'N times the ampere turns of said coils of width (ZYO), where IN is equal to the ampere turns in said outer coil, IN is equal to the ampere turns in said inner coil and said parameters (ZYO), (ZYO), and

are dimensioned within the ranges cross-hatched in FIGS. 6 and 7 of the drawing thereby providing a coil arrangement which will substantially remove a particular quadratic gradient from said volume of magnetic field.

6. Apparatus for improving the homogeneity of a certain volume of magnetic field by removing a certain undesired field gradient including a first and a second pair of substantially rectangular coils, all of said rectangular coils each having substantially the same dimensions including a length which is at least twice its width, said first pair of rectangular coils lying collateral along their lengths in a first plane, said second pair of rectangular coils positioned opposite said first pair in a second plane substantially parallel to said first plane and separated by a distance (ZZO) therefrom, the outer coil sides of the rectangular coils in both planes are separated by a distance (ZYO), the inner coil sides of the rectangular coils in both planes are separated by the distance (ZYO), and said dimensions (Z), (Y0) and (Yo) being selected so that is approximately equal to Y0 2 (zo) +3 Y0 2 (Cz-0) 1 thereby providing a coil arrangement which will substantially remove a certain gradient from said volume of magnetic field.

7. Apparatus for improving the homogeneity of ya certain volume of magnetic field by removing a particular undesired gradient including a first and a second pair of substantially rectangular concentric coils, all of said rectangular coils each having a length which is at least twice as large as its width, said first pair of coils lying in a first plane, said second pair of coils lying opposite to said first pair of coils in a second plane parallel to said first plane and separated by a distance (2Z0) therefrom, said first and second pairs of coils each including an outer coil of width (ZY) and an inner coil of width (ZYO), means for providing each of said coils of width (ZYO) with ampere turns equal to are dimensioned so that is approximately equal to Y0 2 lzti-zti l Y0 2 zo) 3 N 'I' NI is approximately equal to E YO'Y Y0 2 Y0 lntzl 2.] thereby providing a coil arrangement which will substantially remove a particular linear gradient from said volume of magnetic field.

8. Apparatus for improving the homogeneity of a certain volume of magnetic field by removing a certain undesired field gradient including a first and a second pair of substantially rectangular coils, all of said rectangular coils having substantially the same dimensions including a length which is at least twice its width, said first pair of rectangular coils lying collateral along their lengths in a first plane, said second pair of rectangular coils positioned opposite said first pair in a second plane substantially parallel to said first plane and separated by a distance (ZZO) therefrom, the outer coil sides of the rectangular coils in both planes are separated by a distance (ZYO), the inner coil sides of the rectangular coils in both planes are separated by a distance (ZYO) and said dimensions (Z0), (Y0) and (Yo') being selected so that is approximately equal to (1-l-2\/5/5 and yo0 2 (zo) is approximately equal to (1-2\/5/5) thereby providing a coil arrangement which will substantially remove a certain gradient lfrom said volume of magnetic field.

9. Apparatus for improving the homogeneity of a certain volume of magnetic field by removing a certain undesired field gradient including, means for providing a first and second pair of substantially rectangular coils, said rst and second pairs of coils being substantially symmetrical about a point in said certain volume of magnetic field, all of said rectangular coils each having substantially the same dimensions including a length which is at least twice as large as its width (ZYO), said first pair of rectangular coils lying in a first plane and having substantially the same center, the sides along the length of one of said first pair of coils being substantially perpendicular to the sides along the length of the other of said first pair of coils, said secondvpair of coils lying opposite said first pair of coils in a second plane substantially parallel to said first plane and separated therefrom by a distance (ZZO) and the dimensions (ZYD) and (ZZO) being selected so that certain gradient from the volume of magnetic field.

10. Apparatus for improving the homogeneity of a and certain volume of magnetic eld by removing a certain undesired 1ield gradient including, a lirst and second pair -of substantially rectangular spaced apart coils of wire, all of said coils having substantially the same dimensions including lengths which are at least twice as large as their widths, said lirst pair of rectangular coils lying in a rst plane, said second pair of rectangular coils lying in a second plane which is substantially parallel to said first plane so as to dene a gap therebetween, all of said rectangular coils being thin in the direction taken 'across the gap so that said coils lie entirely in said rst and second planes respectively the spacing between said spaced apart coils being larger than the cross section of the wire composing the coils, means to provide opposite directions of current ilow in each of said 15 irst pair of coils, and means to provide opposite directions of current ow in each ,of said second pair of coils so as to provide a coil arrangement which will substantially remove a certain gradient from the volume of magnetic iield.

11. Apparatus according to claim 10 including means lfor supplying a variable direct current to said spaced apart rectangular coils,VJ and means for detecting field Vhomogeneity within the certain volume of magnetic field for sensing maximum field homogeneity.

n 4 References Cited by the Examiner UNITED STATES PATENTS 2,858,504 10/58 Nelson 324-05 9/60 Gunthard 324-05 X 10 Gilbert: Electricity and Magnetism, Revised Edition,

1948, The Macmillan Co., New York, pp. 108 and 109.

Standard Handbook for Electrical Engineers, 8th edition, McGraw-Hill Book Co., N.Y., 1949, Table 4-11 on Y pp. 268 and 269.

Arnold: Physical Review, vol. 102, No. 1, Apr. l, 1956, pp. 136 to 143 incl., principally relied on.

Whiting: Uniform Magnetic Fields, Chalk River, Ontario, February 1957 (CRT-673) (AECL No. 419). Vincent et al.: Nuclear Instruments and Methods, vol.

20 5, No. 4, october 1959, pp. 254 te 25s incl.

25 Examiners.

UNTTED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3,199,021 August 3, 1965 Weston A. Anderson It is hereby certified that error appears in the above numbered patent requiring correction and that the said Letters Patent should read as corrected below.

Column 3, line 2, for "geld" read field line 67, for "eurent" read Current column 4, line 74, for "filed" read field column 5, line 30, for that portion of formula (l) reading I1 D read m=l m=0 line 37, for "1r" read line 55, for cos read cos 2+ same column 5, line 60, for'resigning" read designing column 6, line 22, after "BZ" insert an? equal sign; Column 7 lines 6 to 9 after (Z-ZO)2 l' insert a closing braket; lines 48 and 49, for that portion of the equation reading same Column 7, lines 52 to 54, for that portion of the equation reading (MR2) read (1+R235 same column 7, lines 55 to 57, for that portion of the equation reading B5B -aSB Z d :pirea ay @z2 ay3 azz column 8, line 27, for "Zfield" read field line 62, for

"JX)X, y)=" read JXUC, y)= Column l0, lines 18 and 19,

for

Z :0 read L aYZ w3 same column l0, line 37, for "given" read give lines 74 and 75, for that portion of the equation reading "+0" read +0 column 12, line 34, for

column 14, lines 47 to 49, for that portion of the equation reading column 16, lines 21 and 22, for that portion of the equation reading ('X-ZO) 2 read (z-ZO] Z column 22, lines 44 to 46, for

Signed and sealed this 18th day of October 1966.

(SEAL) Attest:

ERNEST W. SWIDER EDWARD J. BRENNER Attesting Officer Commissioner of Patents 

1. APPARATUS FOR IMPROVING THE HOMOGENEITY OF A CERTAIN VOLUME OF MAGNETIC FIELD BY REMOVING A PARTICULAR UNDESIRED FIELD GRADIENT INCLUDING, ELECTRICALLY CONDUCTIVE MEANS FOR PROVIDING A PLURALITY OF SPACED APART CURRENT PATHS IN EACH OF A PAIR OF ESSENTIALLY PARALLEL PLANES, THE SPACING BETWEEN SAID CURRENT PATHS BEING LARGER THAN ANY DISTANCE TRAVERSING THEIR CROSS SECTION, SAID PAIR OF PLANES POSITIONED ON OPPOSITE SIDES OF THE VOLUME OF MAGNETIC FIELD TO BE CORRECTED SO AS TO DEFINE A GAP THEREBETWEEN, SAID ELECTRICALLY CONDUCTIVE MEANS INE ACH OF SAID PLANES BEING THIN IN THE DIRECTION TAKEN ACROSS THE GAP SO THAT SAID SPACED APART CURRENT PATHS LIE ENTIRELY IN SAID PAIR OF ESSENTIALLY PARALLEL PLANES AND LYING ON OPPOSITE SIDES OF A PLANE THROUGH THE VOLUME OF MAGNETIC FIELD AND PERPENDICULAR TO SAID PARALLEL PLANES, SAID ELECTRICALLY CONDUCTIVE MEANS BEING CONNECTED SO AS TO PROVIDE WHEN ENERGIZED IN EACH OF SAID PARALLEL PLANES TWO OPPOSING DIRECTIONS OF CURRENT FLOW ON ONE SIDE OF SAID PLANE THROUGH SAID VOLUME OF MAGNETIC FIELD AND TWO OPPOSING DIRECTIONS OF CURRENT FLOW ON THE OTHER SIDE OF SAID PLANE THROUGH SAID VOLUME OF MAGNETIC FIELD, AND SAID ELECTRICALLY CONDUCTIVE MEANS BEING DISPOSED SO THAT WHEN ENERGIZED THE CURRENTS FLOWING IN OPPOSING DIRECTIONS WILL FLOW SUBSTANTIALLY PARALLEL TO SAID PLANE THROUGH SAID VOLUME OF MAGNETIC FIELD. 